Collapsing lattice animals and lattice trees in two dimensions

TL;DR

This paper uses advanced simulation methods to analyze phase transitions and universality classes in 2D lattice animals and trees, revealing complex multicritical behavior and critical exponents.

Contribution

It introduces a new sampling technique to precisely study collapse transitions and identifies multiple universality classes, including the Derrida-Herrmann model, in 2D lattice animals.

Findings

Identified a line of second order transitions including a multicritical point at critical percolation.

Determined critical exponents for collapsing trees with high precision.

Provided evidence for a transition between bond-driven and contact-driven collapsed phases.

Abstract

We present high statistics simulations of weighted lattice bond animals and lattice trees on the square lattice, with fugacities for each non-bonded contact and for each bond between two neighbouring monomers. The simulations are performed using a newly developed sequential sampling method with resampling, very similar to the pruned-enriched Rosenbluth method (PERM) used for linear chain polymers. We determine with high precision the line of second order transitions from an extended to a collapsed phase in the resulting 2-dimensional phase diagram. This line includes critical bond percolation as a multicritical point, and we verify that this point divides the line into two different universality classes. One of them corresponds to the collapse driven by contacts and includes the collapse of (weakly embeddable) trees, but the other is {\it not yet} bond driven and does not contain the Derrida-Herrmann model as special point. Instead it ends at a multicritical point $P^*$ where a transition line between two collapsed phases (one bond-driven and the other contact-driven) sparks off. The Derrida-Herrmann model is representative for the bond driven collapse, which then forms the fourth universality class on the transition line (collapsing trees, critical percolation, intermediate regime, and Derrida-Herrmann). We obtain very precise estimates for all critical exponents for collapsing trees. It is already harder to estimate the critical exponents for the intermediate regime. Finally, it is very difficult to obtain with our method good estimates of the critical parameters of the Derrida-Herrmann universality class. As regards the bond-driven to contact-driven transition in the collapsed phase, we have some evidence for its existence and rough location, but no precise estimates of critical exponents.